Comments on A000311 by Tom Copeland, Jan 04 2021


As illustrated in my 2011 formulas here and in A006351, an augmented, signed version of A006351 (1, -1, -2, -8, -52, ...) with elements c_n is the umbral compositional inverse sequence for b_n = A000311(n+1), i.e., (1, 1, 4, 26, 236, ...); that is, (c. + b.)^n = 0^n, the Kronecker delta. This binomial convolution implies that the e.g.f. for the sequence b_n is the reciprocal of that for c_n. For further applicable formulas, see A133314.

In addition, the multiplicative relationship between b_n and c_n given in the 2011 formulas implies that their e.g.f.s are also related via other simple transformations. This supplies information on transformations of the principal branch of the multi-valued Lambert W function, or Euler Tree function, related to the e.g.f.s.

See A134991 for more on the e.g.f.s and the Lambert W function, related to A000169 and A042977, and connections to enumeration of phylogenetic trees, simplicial complexes, projective spaces, and tropical Grassmannians. See A134685, for which this entry is also the row sums, for connections to enumeration of other combinatoric constructs.

The sequences b_n and c_n can be used as the moments for the umbral inverse pair of Appell Sheffer polynomial sequences B_n(x) = (b. + x)^n = Sum_{k=0..n} binomial(n,k) b_k x^{n-k} and C_n(x) = (c. + x)^n, for which the raising operators R defined by, e.g., R B_n(x) = B_{n+1}(x) are given by R = x + d[log(E(t))]/dt with t replaced by d/dx and where E(x) is the e.g.f. of the sequence. Conversely, the raising operators can be used to generate the moments by generating the polynomials through, e.g., R^n 1 = B_n(x) and setting x to zero.